Question 15

Convert the following rational numbers in the form of a terminating decimal or non-terminating and repeating decimal, whichever the case may be, by the process of long division:

(i) $\frac{3}{50}$ (ii) $\frac{2}{9}$

Prove that $\sqrt{5}$ is an irrational number.

Convert the following decimal numbers in the form of $\frac{p}{q}$.

(i) $12.6$ (ii) $0.0120$ (iii) $3.05\bar{2}$ (iv) $1.2\overline{35}$ (v) $0.\overline{23}$ (vi) $2.0\bar{5}$ (vii) $2.12\bar{5}$ (viii) $3.12\bar{5}$ (ix) $2.\overline{1625}$

Locate the following rational numbers on the number line.

(i) $0.532$ (ii) $1.1\bar{5}$

Find 6 rational numbers between 3 and 4.

Find 5 rational numbers between $\frac{2}{5}$ and $\frac{3}{5}$.

Find 5 rational numbers between $\frac{1}{6}$ and $\frac{2}{5}$.

If $\frac{x}{3} + \frac{x}{5} = \frac{16}{15}$, find the rational number $x$.

Let $a$ and $b$ be two non-zero rational numbers such that $a + \frac{1}{b} = 0$. Without assigning any numerical values, determine whether $ab$ is positive or negative. Justify your answer.

A rational number has a terminating decimal expansion whose last non-zero digit occurs in the 4th decimal place. Show that such a number can be written in the form $\frac{p}{10^4}$, where $p$ is an integer not divisible by 10. Is it necessary that the denominator of this rational number, when written in the lowest form, is divisible by $2^4$ or $5^4$? Give reasons.

Without performing division, determine whether the decimal expansion of $\frac{18}{125}$ is terminating or non-terminating. If it terminates, state the number of decimal places.

A rational number in its lowest form has denominator $2^3 \times 5$. How many decimal places will its decimal expansion have? Explain your answer.

Let $a = \frac{7}{12}$ and $b = \frac{5}{6}$. Express both $a$ and $b$ in the form $\frac{k_1}{m}$ and $\frac{k_2}{m}$ where $k_1$, $k_2$ and $m$ are integers and $k_2 - k_1 > 6$. Using the same denominator $m$, write exactly five distinct rational numbers lying between $a$ and $b$ keeping an integer numerator. Explain why the condition $k_2 - k_1 > n + 1$ is necessary to find $n$ such rational numbers between the two rational numbers $a$ and $b$ using this method.

Three rational numbers $x$, $y$, $z$ satisfy $x + y + z = 0$ and $xy + yz + zx = 0$. Show that all the rational numbers $x, y, z$ must be simultaneously zero.

Find the lengths of the hypotenuses of all the right triangles in Fig. 3.14 which is referred to as the square root spiral.